PowerPoint PresentationOverviewThe EconomySecurity StructureSlide 5Arrow-Debreu Security Structure in R2Slide 7Slide 8Arrow-Debreu Security StructureGeneral Security StructureSlide 11Slide 12Slide 13Slide 14Slide 15Slide 16Options to Complete the MarketSlide 18Slide 19Slide 20Vector NotationThree Forms of No-ARBITRAGESlide 23PricingSlide 25State prices qSlide 27The Fundamental Theorem of FinanceMultiple State Prices q & Incomplete MarketsSlide 30Slide 31Slide 32Multiple q in incomplete marketsUniqueness and CompletenessThe Three Asset Pricing FormulasStochastic Discount FactorSlide 37Equivalent Martingale MeasureSlide 39Slide 40Recovering State Prices from Option PricesSlide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Representation of PreferencesAgent’s OptimizationSlide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Pareto EfficiencySlide 60Welfare TheoremsRepresentative Agent & Complete MarketsRepresentative Agent & HARA utility worldSlide 64Extra MaterialPortfolio restrictionsRestricted/Limited ArbitragePortfolio restrictions (ctd.)Slide 69FOR LATER USE Stochastic Discount Factor10:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset PricingLecture 02: One Period ModelLecture 02: One Period ModelProf. Markus K. Brunnermeier10:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset PricingOverviewOverview1.1.Securities StructureSecurities Structure•Arrow-Debreu securities structureArrow-Debreu securities structure•Redundant securitiesRedundant securities•Market completenessMarket completeness•Completing markets with optionsCompleting markets with options2. Pricing (no arbitrage, state prices, SDF, EMM …)3. Optimization and Representative Agent(Pareto efficiency, Welfare Theorems, …)10:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset PricingThe EconomyThe Economy•State space (Evolution of states)Two dates: t=0,1S states of the world at time t=1•PreferencesU(c0, c1, …,cS) (slope of indifference curve)•Security structure Arrow-Debreu economyGeneral security structure0s=1s=2s=S…10:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset PricingSecurity StructureSecurity Structure•Security j is represented by a payoff vector•Security structure is represented by payoff matrix•NB. Most other books use the inverse of X as payoff matrix.10:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset PricingOne A-D asset e1 = (1,0)Payoff Space <X>This payoff cannot be replicated!Arrow-Debreu Security Structure in Arrow-Debreu Security Structure in RR22) Markets are incompleteincompletec1c210:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset PricingAdd secondsecond A-D asset e2 = (0,1) to e1 = (1,0)Arrow-Debreu Security Structure in Arrow-Debreu Security Structure in RR22c1c210:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset PricingArrow-Debreu Security Structure in Arrow-Debreu Security Structure in RR22Payoff space <X>Any payoff can be replicated with two A-D securitiesc1c2Add secondsecond A-D asset e2 = (0,1) to e1 = (1,0)10:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset PricingArrow-Debreu Security Structure in Arrow-Debreu Security Structure in RR22Payoff space <X>New asset is redundantredundant – it does not enlarge the payoff spacec1c2Add secondsecond asset (1,2) to10:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset PricingArrow-Debreu Security StructureArrow-Debreu Security Structure• S Arrow-Debreu securities• each state s can be insured individually • All payoffs are linearly independent• Rank of X = S• Markets are complete10:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset PricingGeneral Security StructureGeneral Security StructureOnly bondPayoff space <X>c1c210:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset PricingGeneral Security StructureGeneral Security StructureOnly bond xbond = (1,1)Payoff space <X>can’t be reachedc1c210:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset PricingAdd security (2,1) to bond (1,1)General Security StructureGeneral Security Structurec1c210:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset Pricing•Portfolio of• buy 3 bonds• sell short 1 risky assetGeneral Security StructureGeneral Security Structurec1c2Add security (2,1) to bond (1,1)10:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset PricingPayoff space <X>Market are complete with security structurePayoff space coincides with payoff space ofGeneral Security StructureGeneral Security StructureTwo asset spanspan the payoff spacec1c210:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset Pricing•Portfolio: vector h 2 RJ (quantity for each asset)•Payoff of Portfolio h is j hj xj = h’X •Asset span<X> is a linear subspace of RS Complete markets <X> = RSComplete markets if and only if rank(X) = SIncomplete markets rank(X) < SSecurity j is redundant if xj = h’X’ with hj=0 General Security StructureGeneral Security Structure10:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset Pricing•Price vector p 2 RJ of asset prices•Cost of portfolio h, •If pj  0 the (gross) return vector of asset j is the vector General Security StructureGeneral Security Structure10:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset PricingOptions to Complete the MarketOptions to Complete the MarketIntroduce call options with final payoff at T:Stock’s payoff:10:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: Asset PricingFin 501: Asset PricingOptions to Complete the MarketOptions to Complete the MarketTogether with the primitive asset we obtainHomework: check whether this markets are complete.10:3210:32 Lecture 02 Lecture 02One Period ModelOne Period ModelFin 501: