# Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Lecture 02: One Period Model Prof. Markus K. Brunnermeier.

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Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Lecture 02: One Period Model Prof. Markus K. Brunnermeier

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Overview 1.Securities Structure Arrow-Debreu securities structureArrow-Debreu securities structure Redundant securitiesRedundant securities Market completenessMarket completeness Completing markets with optionsCompleting markets with options 2.Pricing (no arbitrage, state prices, SDF, EMM …) 3.Optimization and Representative Agent (Pareto efficiency, Welfare Theorems, …)

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model The Economy State space (Evolution of states) Two dates: t=0,1 S states of the world at time t=1 Preferences U(c 0, c 1, …,c S ) (slope of indifference curve) Security structure Arrow-Debreu economy General security structure 0 s=1 s=2 s=S …

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Security 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.

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model One A-D asset e 1 = (1,0) Payoff Space This payoff cannot be replicated! Arrow-Debreu Security Structure in R 2 ) Markets are incomplete c1c1 c2c2

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model second Add second A-D asset e 2 = (0,1) to e 1 = (1,0) Arrow-Debreu Security Structure in R 2 c1c1 c2c2

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Arrow-Debreu Security Structure in R 2 Payoff space Any payoff can be replicated with two A-D securities c1c1 c2c2 second Add second A-D asset e 2 = (0,1) to e 1 = (1,0)

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Arrow-Debreu Security Structure in R 2 Payoff space redundant New asset is redundant – it does not enlarge the payoff space c1c1 c2c2 second Add second asset (1,2) to

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Arrow-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 complete

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model General Security Structure Only bond Payoff space c1c1 c2c2

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model General Security Structure Only bond x bond = (1,1) Payoff space cant be reached c1c1 c2c2

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Add security (2,1) to bond (1,1) General Security Structure c1c1 c2c2

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Portfolio of buy 3 bonds sell short 1 risky asset General Security Structure c1c1 c2c2 Add security (2,1) to bond (1,1)

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Payoff space Market are complete with security structure Payoff space coincides with payoff space of General Security Structure span Two asset span the payoff space c1c1 c2c2

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Portfolio: vector h 2 R J (quantity for each asset) Payoff of Portfolio h is j h j x j = hX Asset span is a linear subspace of R S Complete markets = R S Complete markets if and only if rank(X) = S Incomplete marketsrank(X) < S Security j is redundant if x j = hX with h j =0 General Security Structure

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Price vector p 2 R J of asset prices Cost of portfolio h, If p j 0 the (gross) return vector of asset j is the vector General Security Structure

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Options to Complete the Market Introduce call options with final payoff at T: Stocks payoff:

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Options to Complete the Market Together with the primitive asset we obtain Homework: check whether this markets are complete.

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Overview 1.Securities Structure (AD securities, Redundant securities, completeness, …) 2.Pricing LOOP, No arbitrage and existence of state pricesLOOP, No arbitrage and existence of state prices Market completeness and uniqueness of state pricesMarket completeness and uniqueness of state prices Pricing kernel q *Pricing kernel q * Three pricing formulas (state prices, SDF, EMM)Three pricing formulas (state prices, SDF, EMM) Recovering state prices from optionsRecovering state prices from options 3.Optimization and Representative Agent (Pareto efficiency, Welfare Theorems, …)

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Pricing State space (evolution of states) (Risk) preferences Aggregation over different agents Security structure – prices of traded securities Problem: Difficult to observe risk preferences existence of state pricesWhat can we say about existence of state prices without assuming specific utility functions for all agents in the economy

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Vector Notation Notation: y,x 2 R n y ¸ x, y i ¸ x i for each i=1,…,n. y > x, y ¸ x and y x. y >> x, y i > x i for each i=1,…,n. Inner product y ¢ x = i yx Matrix multiplication

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Three Forms of No-ARBITRAGE 1.Law of one price (LOOP) If hX = kX then p ¢ h = p ¢ k. 2.No strong arbitrage There exists no portfolio h which is a strong arbitrage, that is hX ¸ 0 and p ¢ h < 0. 3.No arbitrage There exists no strong arbitrage nor portfolio k with k X > 0 and p ¢ k · 0.

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Law of one price is equivalent to every portfolio with zero payoff has zero price. No arbitrage ) no strong arbitrage No strong arbitrage ) law of one price Three Forms of No-ARBITRAGE

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Pricing Define for each z 2, If LOOP holds q(z) is a single-valued and linear functional. Conversely, if q is a linear functional defined in then the law of one price holds.

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model LOOP ) q(hX) = p ¢ h A linear functional Q in R S is a valuation function if Q(z) = q(z) for each z 2. Q(z) = q ¢ z for some q 2 R S, where q s = Q(e s ), and e s is the vector with e s s = 1 and e s i = 0 if i s e s is an Arrow-Debreu security q is a vector of state prices Pricing

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model State prices q q is a vector of state prices if p = X q, that is p j = x j ¢ q for each j = 1,…,J If Q(z) = q ¢ z is a valuation functional then q is a vector of state prices Suppose q is a vector of state prices and LOOP holds. Then if z = hX LOOP implies that Q(z) = q ¢ z is a valuation functional, q is a vector of state prices and LOOP holds

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model q1q1 q2q2 p(1,1) = q 1 + q 2 p(2,1) = 2q 1 + q 2 Value of portfolio (1,2) 3p(1,1) – p(2,1) = 3q 1 +3q 2 -2q 1 -q 2 = q 1 + 2q 2 State prices q c1c1 c2c2

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model The Fundamental Theorem of Finance Proposition 1. Security prices exclude arbitrage if and only if there exists a valuation functional with q >> 0. Proposition 2. Let X be an J S matrix, and p 2 R J. There is no h in R J satisfying h ¢ p · 0, h X ¸ 0 and at least one strict inequality if, and only if, there exists a vector q 2 R S with q >> 0 and p = X q. No arbitrage, positive state prices

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Multiple State Prices q & Incomplete Markets q1q1 q2q2 c1c1 c2c2 p(1,1) Payoff space bond (1,1) only What state prices are consistent with p(1,1)? p(1,1) = q 1 + q 2 = One equation – two unknowns q 1, q 2 There are (infinitely) many. e.g. if p(1,1)=.9 q 1 =.45, q 2 =.45 or q 1 =.35, q 2 =.55

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model q q complete markets

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model q q p=Xq incomplete markets

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model q qoqo p=Xq o incomplete markets

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Multiple q in incomplete markets qv q* qoqo p=Xq Many possible state price vectors s.t. p=Xq. One is special: q* - it can be replicated as a portfolio. c2c2 c1c1

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Uniqueness and Completeness uniqueProposition 3. If markets are complete, under no arbitrage there exists a unique valuation functional. If markets are not complete, then there exists v 2 R S with 0 = Xv. Suppose there is no arbitrage and let q >> 0 be a vector of state prices. Then q + v >> 0 provided is small enough, and p = X (q + v). Hence, there are an infinite number of strictly positive state prices.

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model The Three Asset Pricing Formulas State pricesState pricesp j = s q s x s j Stochastic discount factorStochastic discount factorp j = E[mx j ] Martingale measureMartingale measurep j = 1/(1+r f ) E [x j ] (reflect risk aversion by over(under)weighing the bad(good) states!) m1m1 m2m2 m3m3 xj1xj1 xj2xj2 xj3xj3 ^

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Stochastic Discount Factor That is, stochastic discount factor m s = q s / s for all s.

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model c1c1 Stochastic Discount Factor shrink axes by factor m m*

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Price of any asset Price of a bond Equivalent Martingale Measure

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model The Three Asset Pricing Formulas State pricesState pricesp j = s q s x s j Stochastic discount factorStochastic discount factorp j = E[mx j ] Martingale measureMartingale measurep j = 1/(1+r f ) E [x j ] (reflect risk aversion by over(under)weighing the bad(good) states!) m1m1 m2m2 m3m3 xj1xj1 xj2xj2 xj3xj3 ^

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model specify Preferences & Technology observe/specify existing Asset Prices State Prices q (or stochastic discount factor/Martingale measure) derive Asset Prices derive Price for (new) asset evolution of states risk preferences aggregation absolute asset pricing relative asset pricing NAC/LOOP LOOP NAC/LOOP Only works as long as market completeness doesnt change

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Suppose that S T, the price of the underlying portfolio (we may think of it as a proxy for price of market portfolio), assumes a "continuum" of possible values. Suppose there are a continuum of call options with different strike/exercise prices ) markets are complete Let us construct the following portfolio: for some small positive number >0, Buy one call with Sell one call with Buy one call with. Recovering State Prices from Option Prices

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Figure 8-2 Payoff Diagram: Portfolio of Options Recovering State Prices … (ctd.)

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Let us thus consider buying 1 / units of the portfolio. The total payment, when, is, for any choice of. We want to let, so as to eliminate the payments in the ranges and.The value of 1 / units of this portfolio is : Recovering State Prices … (ctd.)

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Taking the limit e ! 0

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model The value today of this cash flow is : Recovering State Prices … (ctd.) Evaluating following cash flow

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model specify Preferences & Technology observe/specify existing Asset Prices State Prices q (or stochastic discount factor/Martingale measure) derive Asset Prices derive Price for (new) asset evolution of states risk preferences aggregation absolute asset pricing relative asset pricing NAC/LOOP LOOP NAC/LOOP Only works as long as market completeness doesnt change

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Overview 1.Securities Structure (AD securities, Redundant securities, completeness, …) 2.Pricing (no arbitrage, state prices, SDF, EMM …) 3.Optimization and Representative Agent Marginal Rate of Substitution (MRS)Marginal Rate of Substitution (MRS) Pareto EfficiencyPareto Efficiency Welfare TheoremsWelfare Theorems Representative Agent EconomyRepresentative Agent Economy

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Representation of Preferences A preference ordering is (i) complete, (ii) transitive, (iii) continuous [and (iv) relatively stable] can be represented by a utility function, i.e. (c 0,c 1,…,c S ) Â (c 0,c 1,…,c S ), U(c 0,c 1,…,c S ) > U(c 0,c 1,…,c S ) (more on risk preferences in next lecture)

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Agents Optimization Consumption vector (c 0, c 1 ) 2 R + x R + S Agent i has U i : R + x R + S ! R endowments (e 0,e 1 ) 2 R + x R + S U i is quasiconcave {c: U i (c) ¸ v} is convex for each real v U i is concave: for each 0 ¸ ¸ 1, U i ( c + (1- )c) ¸ U i (c) + (1- ) U i (c) U i / c 0 > 0, U i / c 1 >>0

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Agents Optimization Portfolio consumption problem

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Agents Optimization For time separable utility function and expected utility function (later more)

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Agents Optimization To sum up Proposition 3: Suppose c* >> 0 solves problem. Then there exists positive real numbers, 1, …, S, such that The converse is also true. The vector of marginal rate of substitutions MRS s,0 is a (positive) state price vector.

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Mr. A cA0cA0 cAscAs Suppose c s is fixed since it cant be traded {c A 0, c A 1 :U(c A 0, c A 1 )=U(e)} e slope As indifference curve

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Ms. B cBscBs cB0cB0 e

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Ms. B cBscBs cB0cB0 Mr. A cAscAs cA0cA0 e A, e B

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Ms. B cBscBs cB0cB0 Mr. A cAscAs cA0cA0 e A, e B 1 q

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Ms. B cBscBs cB0cB0 Mr. A cAscAs cA0cA0 e A, e B Set of PO allocations (contract curve)

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Pareto Efficiency Allocation of resources such that there is no possible redistribution such that at least one person can be made better off without making somebody else worse off Note Allocative efficiency Informational efficiency Allocative efficiency fairness

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Ms. B cBscBs cB0cB0 Mr. A cAscAs cA0cA0 e A, e B Set of PO allocations (contract curve) MRS A = MRS B

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Welfare Theorems First Welfare Theorem. If markets are complete, then the equilibrium allocation is Pareto optimal. State price is unique q. All MRS i (c*) coincide with unique state price q. Second Welfare Theorem. Any Pareto efficient allocation can be decentralized as a competitive equilibrium.

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Representative Agent & Complete Markets Aggregation Theorem 1: Suppose markets are complete many agents single agent/planner Then asset prices in economy with many agents are identical to an economy with a single agent/planner whose utility is U(c) = k k u k (c), where k is the welfare weights of agent k. and the single agent consumes the aggregate endowment.

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Representative Agent & HARA utility world Aggregation Theorem 2: Suppose riskless annuity and endowments are tradeable. agents have common beliefs agents have a common rate of time preference agents have LRT (HARA) preferences with R A (c) =1/(A i +Bc ) ) linear risk sharing rule many agents single agent Then asset prices in economy with many agents are indentical to a single agent economy with HARA preferences with R A (c) = 1/( i A i + B).

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Overview 1.Securities Structure (AD securities, Redundant securities, completeness, …) 2.Pricing (no arbitrage, state prices, SDF, EMM …) 3.Optimization and Representative Agent (Pareto efficiency, Welfare Theorems, …)

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Extra Material Follows!

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Portfolio restrictions Suppose that there is a short-sale restriction where is a convex set d is a convex set F or z 2 let (cheapest portfolio replicating z)

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Restricted/Limited Arbitrage An arbitrage is limited if it involves a short position in a security In the presence of short-sale restrictions, security prices exclude (unlimited) arbitrage (payoff 1 ) if, and only if, here exists a q >> 0 such that Intuition: q = MRS^i from optimization problem some agents wished they could short-sell asset

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Portfolio restrictions (ctd.) As before, we may define R f = 1 / s q s, and s can be interpreted as risk-neutral probabilities R f p j ¸ E [x j ], with = if 1/R f is the price of a risk-free security that is not subject to short-sale constraint.

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model Portfolio restrictions (ctd.) Portfolio consumption problem max over c 0, c 1, h utility function u(c 0,c 1 ) subject to (i) 0 · c 0 · w 0 – p ¢ h (ii) 0 · c 1 · w 1 + h X Proposition 4: Suppose c*>>0 solves problem s.t. h j ¸ –b j for. Then there exists positive real numbers, 1, 2,..., S, such that – U i / c 0 (c*) = – U i / c 0 (c*) = ( 1,..., S ) – s The converse is also true.

Fin 501: Asset Pricing 19:48 Lecture 02One Period Model FOR LATER USE Stochastic Discount Factor That is, stochastic discount factor m s = q s / s for all s.

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